1.e4 or 1.d4?

1.e4 is likely to lead to open and semi-open games, while 1.d4 is more
likely to lead to closed and other more murky structures.

By 'open game' I mean a position after the opening which features open
files and diagonals, a fairly clear centre and no locked pawn chains.
Closed games are the opposite.

Hmm, why is that?

After 1.e4 e5, White can follow up with d2-d4, immediately or after some
preparation. White gets open diagonals for the Bishops, and (at least)
the chance to get (at least) a half-open file in the centre.

After 1.d4 d5, e2-e4 looks like it loses a pawn right now, and may be hard to
organise later. If White has to settle for e2-e3, then White may be obliged to
develop the Bc1 before touching the Bf1 and castling. So, the game
unfolds more slowly, and open files and diagonals emerge only later.

After 1.d4 Nf6, White may even be invited to play e2-e4, but where any
open lines will emerge is not clear.

Why should you care?

Well, open games tend to be easier to follow, easier to play, and are
easier settings in which to practice the sorts of things you should be
practising when starting out (fast development, king's-side attacks,
tactics, endgames). Closed and Indian systems are not as easy to play or
understand. You have to put up with Black's other responses to 1.e4, but
against at least some of them you can steer for an IQP position. If you
learn how to play IQP positions, you may not always be able to claim an
opening advantage, but you will always be able to know what you are doing.

So, if you start with 1.e4, you're making a good choice, and one that
you needn't ever change. The Italian Game is a good place to start,
while the Ruy Lopez and Scotch Game are both still played by
Grandmasters.

1.e3 or 1.d3 or neither?

Short answer: neither.

Long answer: Either or both!

An opening is only as good as the ideas you bring to it. (I think that's
a quote, but I Googled it, and got this website... Could someone let me
know where I stole it from?)

If you make moves at random, you might hit on 1.e3 d5 2.a4, or 1.d3 d5
2.Qd2, but those are not opening systems. You have to have a set of
ideas you can bring to oppose different Black approaches. You can play
either first move with a ready-made set of ideas, maybe e3/b3/Bb2/Bb4
(Nimzo-Larsen Attack), or d3/g3/Bg2 (King's Indian Attack), and do well.
You may note that two of our strongest players, Graham Bolt and Tim
Paulden, have been using the Hippopotamus with both White and Black,
which uses both e3 and d3, and doing pretty well.

King's Indian Defence or Grunfeld?

Short answer: King's Indian.

Long answer:

In the King's Indian Defence, White is invited to set up a big pawn
centre with c4, d4 and e4.
1.d4 Nf6 2.c4 g6 3.Nc3 d6 4.e4 Bg7 5.Nf3 0-0
Black generally hits back in the centre with
....e7-e5, when one likely result is that White pushes on with d4-d5,
resulting in blocked central pawn chains, rather like the French
Defence.
6.Be2 e5
By the way, Black doesn't lose the pawn on e5:
7.dxe5 dxe5 8.Nxe5 Qxd1+ 9.Bxd1 Nxe4!7.0-0 Nc6 8.d5 Ne7

I've already talked about the virtues of rehearsing a particular
pawn structure, the IQP, above. Similarly, a KID player gets to play
this blocked centre position repeatedly, and will be able to adopt and
adapt from known plans and piece placements. Black's pawns point to more
space on the King's-side and a break with ...f5; White has more
Queen's-side space and a break with c4-c5.

Viktor Korchnoi was always known as a King's Indian Killer, being able
to show the virtues of the White structure against the strongest players
in the world. But this is how he got on against Bobby Fischer in a
rapidplay game:

Black is doing OK in all of these structures and all these variations,
but my guess is that you are going to have to understand more different
structures than in the KID, and you are going to have to rely more on
exact knowledge than general plans and analogies from similar positions.

Benoni: immediate or delayed?

Short answer: delayed, but try the Benko instead.

Long answer:

The classic Modern Benoni structure is one where the positions are
lively and unbalanced, and where the overall plans for each side are
clear enough but the complications may be intense.

Black has a Queen's-side majority, White a central majority. Black may
advance on the Queen's-side, supported by the strong Bg7 on the long
diagonal. White is going to try and push through e4-e5. White can try
and do this with pieces alone, but it's natural to seek support by f2-
f4. So, White might move the Knight from f3 to... hmm, where, exactly?
Well, how about d2, thence to c4? On c4, it supports e4-e5, and if
Black is restrained from ...b5 by a2-a4, it will stand very well.

So, Black players devised a cunning plan: delaying ...e6xd5. Black
reasons that White is going to run out of constructive moves. Black can
prepare ...b5 for a long while without taking on d5, while White simply
cannot play Nf3-d2-c4, because there is a pawn on c4.

This is one argument for avoiding the Schmidt Benoni which uses 1.d4 c5
2.d5. White gets a very early signal about the structure, and can delay
or avoid c2-c4, keeping c4 free for a piece.

Overall, I might prefer the Benko for Black: you get a small range of
structures and the initiative. I have got into the habit recently of
describing the art of chess as creating a position which is too hard for
your opponent but not you. I think the Benko is easier to play for Black
than for White.